I tested it out with a USB-C charger, a power meter tester, and one of these docks that I bought about a year ago. It does work, but fast charging (power delivery) is not enabled when going through the dock.

I use my iPhone 6+ on my desk a lot and sometimes want a private call with headphones rather than through the mac. As i am thinking of upgrading to iPhone 7+, does this allow iPhone 7 to charge and also plug in headphones for making phone calls?

It will allow those cases to fit just fine.If you can see the little white circle around the lighting connector, thats to create a gap between your phone and the dock which gives enough space for your case and phone to sit in the right place.

No, the dock is simply a "middle-man" between the Lightning cord and the charging slot on the bottom of your device. You must plug the dock into the wall using your Lightnig cable, and then dock your iPhone into the dock. The dock itself does not store any actual battery power.